Question

How do you do this Question?​

Answer 1

Answer:

$a = \frac{ - 13}{3}$

Step-by-step explanation:

Let the first polynomial be p(x) and the second polynomial be g(x)

$p(x) = {2x}^{3} + {ax}^{2} + 3x - 5$

$g(x) = {x}^{3} + {x}^{2} - 4x + a$

Since x-2=0

x=0+2

x=2

So, solving p(x)

$p(2) = {2(2)}^{3} + {a(2)}^{2} + 3(2) - 5$

$p(2) = {2}^{4} + 4a + 6 - 5$

$p(2) = 16 + 4a + 6 - 5$

$p(2) = 17 + 4a \: (1)$Now, solving g(x)

$g(x) = {2}^{3} + {2}^{2} - 4(2) + a$

$g(x) = 8 + 4 - 8 + a$

$g(x) = 4 + a \: (2)$

Remainders (1) and (2) are of p(x) and g(x) respectively

But ATQ,

(1)=(2)

$17 + 4a = 4 + a$

$4a - a = 4 - 17$

$3a = - 13$

So, the answer is

$a = \frac{ - 13}{3}$